MOSER’S THEOREM ON MANIFOLDS WITH CORNERS
MARTINS BRUVERIS, PETER W. MICHOR, ADAM PARUSIŃSKI, ARMIN RAINER
Abstract. Moser’s theorem [13] states that the diffeomorphism group of a
compact manifold acts transitively on the space of all smooth positive densities
with fixed volume. Here we describe the extension of this result to manifolds
with corners. In particular we obtain Moser’s theorem on simplices. The proof
is based on Banyaga’s paper [1], where Moser’s theorem is proven for manifolds
with boundary. A cohomological interpretation of Banyaga’s operator is given,
which allows a proof of Lefschetz duality using differential forms.
1. Introduction. In [13] Moser proved that on a connected compact oriented manifold M
R without
R boundary there exists for any two positive volume forms µ0 and µ1
with M µ0 = M µ1 an orientation preserving diffeomorphism ϕ with ϕ∗ µ1 = µ0 .
In [1] Banyaga extended this to compact oriented manifolds with boundary and
showed that the diffeomorphism can be chosen such that it restricts to the identity
on the boundary. On a manifold with corners one cannot expect the diffeomorphism to be the identity on the boundary, since at a corner x of index 2 or higher
the derivative of such a diffeomorphism would have to be the identity: in this case
x lies in the boundary of at least two codimension 1 strata of ∂M . So the derivative
at x restricted to two codimension 1 subspaces is the identity and thus it has to
be the identity on the whole space; in particular the Jacobian determinant there
equals 1.
Moser’s theorem on manifolds with corners is needed for example in [2]. Even on
simplices it does not seem to be known, but is highly desirable. In fact, Banyaga’s
method [1] gives just the desired result. But this is not immediately obvious and
it took us a long time to realize it. Therefore we think that it is worth wile to
write the proof with all details. Along the way we also prove Stokes’ theorem on
manifolds with corners.
For related results see [5] for a version of Moser’s theorem on bounded domains
in Rm with low differentiability requirements furnishing diffeomorphisms with only
low regularity using PDE techniques; this does not imply the result given here. See
also the recent book [4] for results on k-forms instead of volume forms. A version
on non-compact manifolds is in [8] which also sketches a proof for non-compact
manfolds with boundary which differs from Banyaga’s proof.
Date: January 4, 2017.
2010 Mathematics Subject Classification. Primary 53C65, 58A10.
Key words and phrases. manifolds with corners, Moser’s theorem, Stokes’ theorem.
Supported by the Austrian Science Fund (FWF), Grant P 26735-N25, and by a BRIEF Award
from Brunel University London.
1
2
MARTINS BRUVERIS, PETER W. MICHOR, ADAM PARUSIŃSKI, ARMIN RAINER
2. Manifolds with corners alias quadrantic (orthantic) manifolds. For
more information we refer to [7], [11], [10]. Let Q = Qm = Rm
≥0 be the positive orthant or quadrant. By Whitney’s extension theorem or Seeley’s theorem,
the restriction C ∞ (Rm ) → C ∞ (Q) is a surjective continuous linear mapping which
admits a continuous linear section (extension mapping); so C ∞ (Q) is a direct summand in C ∞ (Rm ). A point x ∈ Q is called a corner of codimension (or index)
q > 0 if x lies in the intersection of q distinct coordinate hyperplanes. Let ∂ q Q
denote the set of all corners of codimension q.
A manifold with corners (recently also called a quadrantic manifold) M is a
smooth manifold modeled on open subsets of Qm . We assume that it is connected
and second countable; then it is paracompact and each open cover admits a subordinated smooth partition of unity. Any manifold with corners M is a submanifold
with corners of an open manifold M̃ of the same dimension, and each smooth function on M extends to a smooth function on M̃ . We do not assume that M is
oriented, but for Moser’s theorem we will eventually assume that M is compact.
Let ∂ q M denote the set of all corners of codimension q. Then ∂ q M is a submanifold
without boundary of codimension q in M ; it has finitely many connected components if M is compact. We shall consider ∂M as stratified into the connected
components of all ∂ q M for q > 0. Abusing notation we will call ∂ q M the boundary
stratum of codimension q; this will lead to no confusion. Note that ∂M itself is not
a manifold with corners. We shall denote by j∂ q M : ∂ q M → M the embedding of
the boundary stratum of codimension q into M , and by j∂M : ∂M → M the whole
complex of embeddings of all strata.
Each diffeomorphism of M restricts to a diffeomorphism of ∂M and to a diffeomorphism of each stratum ∂ q M . The Lie algebra of Diff(M ) consists of all vector
fields X on M such that X|∂ q M is tangent to ∂ q M . We shall denote this Lie
algebra by X(M, ∂M ).
3. Differential forms. There are several differential complexes on a manifold with
corners. If M is not compact there are also the versions with compact support.
• Differential forms that vanish near ∂M . If M is compact, this is the same
as the differential complex Ωc (M \ ∂M ) of differential forms with compact
support in the open interior M \ ∂M .
• Ω(M, ∂M ) = {α ∈ Ω(M ) : j∂∗q M α = 0 for all q ≥ 1}, the complex of
differential forms that pull back to 0 on each boundary stratum.
• Ω(M ), the complex of all differential forms. Its cohomology equals singular
cohomology with real coefficients of M , since R → Ω0 → Ω1 → . . . is a
fine resolution of the constant sheaf on M ; for that one needs existence of
smooth partitions of unity and the Poincaré lemma which holds on manifolds with corners. The Poincaré lemma can be proved as in [12, 9.10] in
each quadrant.
If M is an oriented manifold with corners of dimension m and if µ ∈ Ωm (M ) is a
nowhere vanishing form of top degree, then X(M ) 3 X 7→ iX µ ∈ Ωm−1 (M ) is a
linear isomorphism. Moreover, X ∈ X(M, ∂M ) (tangent to the boundary) if and
only if iX µ ∈ Ωm−1 (M, ∂M ).
3
4. Towards the long exact sequence of the pair (M, ∂M ). Let us consider
the short exact sequence of differential graded algebras
0 → Ω(M, ∂M ) → Ω(M ) → Ω(M )/Ω(M, ∂M ) → 0 .
The complex Ω(M )/Ω(M, ∂M ) is a subcomplex of the product of Ω(N ) for all
connected components N of all ∂ q M . The quotient consists of forms which extend
continuously over boundaries to ∂M with its induced topology in such a way that
one can extend them to smooth forms on M ; this is contained in the space of
‘stratified forms’ as used in [15]. There Stokes’ formula is proved for stratified
forms.
5. Proposition (Stokes’ theorem). For a connected oriented manifold M with
corners of dimension dim(M ) = m and for any ω ∈ Ωm−1
(M ) we have
c
Z
Z
dω =
j∂∗1 M ω .
M
∂1M
Note that ∂ 1 M may have several components. Some of these might be noncompact.
We shall deduce this result from Stokes’ formula for a manifold with boundary by
making precise the fact that ∂ ≥2 M has codimension 2 in M and has codimension
1 with respect to ∂ 1 M . The proof also works for manifolds with more general
boundary strata, like manifolds with cone-like singularities. A lengthy full proof
can be found in [3].
Proof. We first choose a smooth Rdecreasing function f on R≥0 such that f = 1 near
∞
0 and f (r) = 0 for r ≥ ε. Then 0 f (r)dr < ε and for Qm = Rm
≥0 with m ≥ 2,
Z
Z ∞
Z ∞
f 0 (|x|) dx = Cm f 0 (r)rm−1 dr = Cm f (r)(rm−1 )0 dr
m
Q
0
0
Z ε
= Cm
f (r)(rm−1 )0 dr ≤ Cm εm−1 ,
0
where Cm denotes the surface area of S m−1 ∩ Qm . Given ω ∈ Ωm−1
(M ) we use
c
the function f Son quadrant charts on M to construct a function gRon M that
is 1
near ∂ ≥2 M = q≥2 ∂ q M , has support close to ∂ ≥2 M and satisfies M dg ∧ ω < ε.
Then (1 − g)ω is an (m − 1)-form with compact support in the manifold with
boundary M \ ∂ ≥2 M , and Stokes’ formula (cf. [12, 10.11]) now says
Z
Z
d((1 − g)ω) =
j∂∗1 M ((1 − g)ω) .
M \∂ ≥2 M
∂1M
But ∂ ≥2 M is a null set in M and the quantities
Z
Z
Z
Z
∗
d((1
−
g)ω)
−
dω
and
j
((1
−
g)ω)
−
∂1M
M
M
are small if ε is small enough.
∂1M
∂1M
j∂∗1 M ω 6. Theorem (Moser’s theorem for manifolds with corners). Let M be a compact
connected smooth manifold with corners, possibly
non-orientable.
Let µ0 , µ1 ∈
R
R
Dens+ (M ) be smooth positive densities with M µ0 = M µ1 . Then there exists
4
MARTINS BRUVERIS, PETER W. MICHOR, ADAM PARUSIŃSKI, ARMIN RAINER
a diffeomorphism ϕ : M → M such that µ1 = ϕ∗ µ0 . Moreover, ϕ can be chosen to
be the identity on ∂M if and only if µ0 = µ1 on ∂ ≥2 M .
Proof. We first prove the theorem for oriented M . In this case Dens+ (M ) equals
the space Ωm
+ (M ) of positive m-forms for m = dim(M ). Put µt := µ0 + t(µ1 − µ0 )
for t ∈ [0, 1]; then each µt is a volume form on M since these form a convex set.
We look for a curve of diffeomorphisms,
t 7→ ϕt , with ϕ∗t µt = µ0 ; this curve has to
R
∂
∗
(µ1 − µ0 ) = 0, we have [µ1 − µ0 ] = 0 ∈ H m (M ). Fix
satisfy ∂t (ϕt µt ) = 0. Since
M
R
m
ω ∈ Ωc (M \ ∂M ) with ω = 1. Using lemma 7 below we have
ψ := I ω (µ1 − µ0 ) ∈ Ωm−1 (M, ∂M ) with
Z
dψ = dI ω (µ1 − µ0 ) = µ1 − µ0 − ω
(µ1 − µ0 ) = µ1 − µ0 .
M
Put ηt :=
∂
ϕt ) ◦ ϕ−1
( ∂t
t ;
then by well known formulas (see [12, 31.11], e.g.) we have:
wish ∂
∗
∂t (ϕt µt )
0 =
∂
= ϕ∗t Lηt µt + ϕ∗t ∂t
µt = ϕ∗t (Lηt µt + µ1 − µ0 ) ,
wish
0 = Lηt µt + µ1 − µ0 = diηt µt + iηt dµt + dψ = diηt µt + dψ .
We can choose ηt uniquely by requiring that iηt µt = −ψ, since µt is non-degenerate
for all t. The time dependent vector field ηt is tangent to each boundary stratum
∂ q M , since ψ ∈ Ω(M, ∂M ). Then the evolution operator ϕt = Φηt,0 exists for
t ∈ [0, 1] since M is compact, by [12, 3.30]. Moreover, ϕt : ∂M → ∂M . Thus ϕt
restricts to a diffeomorphism of M for each t. On M we have, using [12, 31.11.2],
∗
∂
∂t (ϕt µt )
= ϕ∗t (Lηt µt + dψ) = ϕ∗t (diηt µt + dψ) = 0 ,
so ϕ∗t µt = constant = µ0 . If µ0 = µ1 on ∂ ≥2 M , then ψ = I ω (µ1 − µ0 ) vanishes
along ∂M by lemma 7 and hence so does ηt , thus ϕt is the identity there.
If M is not orientable, we let p : or(M ) → M be the 2-sheeted orientable
double cover of M : It is the Z2 -principal bundle with cocycle of transition functions
sign det d(uβ ◦ u−1
α )(uα (x)) where (Uα , uα ) is a smooth atlas for M . Each connected
(thus orientable) chart of M appears twice as chart of or(M ), once with each
orientation. Thus or(M ) is again a smooth manifold with corners. Let τ : or(M ) →
or(M ) be the orientation reversing deck-transformation; see [12, 13.1]. Pullback
∗
p∗ : Ω(M ) → Ω(or(M )) is an isomorphism onto the eigenspace Ω(or(M ))τ =1 of τ ∗
with eigenvalue 1. The space Dens+ (M ) of positive smooth densities on M is via
∗
p∗ isomorphic to the space of positive m-forms in the eigenspace Ωm (or(M ))τ =−1
∗
of τ with eigenvalue −1; these are the ‘formes impaires’ of de Rham. Note the
abuse of notation here: p∗ of a density differs (by local signs) from p∗ of a form.
See [12, 13.1 and 13.3] for more details.
We consider the pullback densities p∗ µt as positive m-forms denoted
R by νt on
or(M ) which satisfy τ ∗ νt = −νt , and for ω ∈ Ωm
(or(M
)\∂
or(M
))
with
ω=1
c
or(M )
we choose
ψ̃ = I ω (ν1 − ν0 ) ∈ Ωm−1 (or(M ), ∂ or(M )) which satisfies
Z
dψ̃ = ν1 − ν0 − ω ·
(ν1 − ν0 ) = ν1 − ν0 .
or(M )
5
Let ψ = 12 ψ̃ − 12 τ ∗ ψ̃, then again dψ = ν1 − ν0 and now also τ ∗ ψ = −ψ. A timedependent vector field ηt is uniquely given by iηt νt = −ψ.
iτ ∗ ηt νt = −iτ ∗ ηt τ ∗ νt = −τ ∗ (iηt νt ) = τ ∗ ψ = −ψ
=⇒
τ ∗ ηt = ηt .
In particular, the time dependent vector field ηt is tangent to each boundary stratum
∂ q or(M ), and it projects to a time dependent vector field on M whose evolution
gives the curve of diffeomorphisms with all required properties.
7. Lemma. Let M be an oriented connected manifold with
R corners of dimension
dim(M ) = m. For each form ω ∈ Ωm
ω = 1 there exists a
c (M \ ∂M ) with
continuous linear operator
m−1
I ω : Ωm
(M, ∂M ) such that:
c (M ) → Ωc
R
• d I ω (α) = α − ω α for all α ∈ Ωm
c (M ).
≥2
ω
• If α vanishes on ∂ M then I (α) vanishes along ∂M .
For a compact oriented manifold with boundary, this is due to Banyaga [1]. We
call I ω the Banyaga operator.
ω
Proof. We first construct Im
for the case when M is a partial quadrant Qm
p :=
p
m−p
m
1
R≥0 × R
= {x ∈ R : x ≥ 0, . . . , xp ≥ 0}.
ω
by induction on the dimension m and start with I0ω = 0.
We construct Im
R
For Q = Q11 = R≥0 we consider ω = g(u)du with supp(g) ⊂ R>0 and g(u)du =
1. Then for α = a(u)du ∈ Ω1c (Q) we put
Z u
Z
I1ω (α)(u) : =
a(t) − g(t)
a(v)dv dt
so that
0
Q
Z
dI1ω (α) = α − ω α and I1ω (α)(0) = 0 ,
and thus I1ω (α) ∈ Ω0 (R≥0 , {0}). For Q = Q10 = R we just integrate from −∞ to u.
Note that I1ω (α)(u) vanishes for large u, so it has compact support.
p
m−p
For general Q = Qm
we start with a specific form ω: Let g(t) be
p = R≥0 × R
R
a smooth function on R with compact support in R>0 and g(t) dt = 1. Then we
put
ω = g(u1 )du1 ∧ · · · ∧ g(um )dum =: ω 0 ∧ g(um )dum ,
d : Ωm−1 (Q) → Ωm (Q),
d=
m−1
X
dui ∧ ∂ui + dum ∧ ∂um =: d0 + dum ∧ ∂um .
i=1
Any form α ∈
m
Ωm
c (Qp )
(
α1 :
can be written as α = α1 (um ) ∧ dum for a smooth curve
(Qm−1
)
R → Ωm−1
c
p
m−1
m−1
R≥0 → Ωc (Qp−1 )
if
if
0 ≤ p ≤ m − 1,
p = m.
Following an idea of de Rham [6] used by [1], we put
0
ω
ω
Im
(α) = Im−1
(α1 (um )) ∧ dum +
6
MARTINS BRUVERIS, PETER W. MICHOR, ADAM PARUSIŃSKI, ARMIN RAINER
R m R
R
 u
dt
m−1 α1 (t) − g(t)
m α
−∞
Q
Q
+ (−1)m−1 ω 0 · R um R p
R p 
dt
m−1 α1 (t) − g(t)
m α
0
Qp−1
Qp
if
0 ≤ p ≤ m − 1,
if
p = m,
m
which is in Ωm−1 (Qm
p , ∂Qp ): The first summand by induction on m and because
m
it contains du . The second summand since the integral starts at 0 in the relevant
ω
case. Im
(α) has compact support: The first summand by induction, and the second
summand since ω 0 has compact support in the first m − 1 variables, and since the
integral vanishes for large um . If α vanishes on ∂ ≥2 Qm then α1 (um ) vanishes on
∂ ≥2 Qm−1 in both cases. Then the first summand vanishes on ∂Qm by induction,
and second summand since the integral starts at 0 in the relevant case.
ω
Next we compute the exterior derivative of Im
(α). For the first summand
0
0
ω
m
m
0 ω
m
d Im−1 (α1 (u )) ∧ du = d Im−1 (α1 (u )) ∧ dum =
Z
= α1 (um ) − ω 0
α1 (um ) ∧ dum by induction
Qm−1
Z
= α − (ω 0 ∧ dum )
α1 (um ) .
Qm−1
The exterior derivative of the second summand is
Z
Z
m−1
m
0
m
m
(−1)
du ∧ ω
α1 (u ) − g(u )
α
Qm−1
Qm
Z
Z
0
m
m
0
m
m
= (ω ∧ du )
α1 (u ) − (ω ∧ g(u )du )
Qm−1
ω
which proves dIm
(α) = α − ω
R
Qm
α
Qm
α.
m
m
In order to change to another m-form ω̃ ∈ Ωm
c (Qp \ ∂Qp ) with
put
Z
ω̃
ω
ω
Im (α) = Im (α) − I (ω̃)
α.
R
Qm
p
ω̃ = 1 we
Qm
p
ω
to the oriented manifold with corners M . We
Now we extend the operators Im
construct an oriented atlas similarly to [1, lemme 1] with the property that all charts
contain a common chart U0 . Choose x0 ∈ M \ ∂M and a closed neighborhood V0
of x0 in M \ ∂M which is diffeomorphic to a closed ball in Rm . For each y ∈ M \ V0
choose an oriented open chart ϕy : Uy → Qm
py centered at y onto some partial
quadrant, xy ∈ Uy \ ∂Uy and a smooth embedded curve cy in M \ ∂M from xy to
x0 . Then choose a vector field Xy with Xy (cy (t)) = c0y (t) for each t that vanishes
X
at y and on ∂M . The flow Flt y moves xy along cy to x0 and keeps y and ∂M fixed.
X
Fl1 y also maps an open neighborhood of xy in Uy to an open neighborhood of x0 ,
which we may extend to an open neighborhood of V0 via a diffeomorphism ψy of
M that is the identity near y and near ∂M . Now consider the charts
X
ψy−1
ψy (Fl1 y (Uy )) −
X
X
Fl−1y
→ Fl1 y (Uy ) −
ϕ
→ Uy − y→ Qm
py
and call the resulting atlas again (Uy , ϕy ). We choose a smooth partition of unity
λy with a locally finite family of supports subordinated
T to this atlas (most of the
λy are 0). Finally we choose the chart (U0 , ϕ0 ) inside y Uy which is possible since
the intersection contains the neighborhood V0 .
7
R
Choose ω ∈ Ωm
c (U0 ) with M ω = 1 and let
X
(ϕ−1 )∗ ω
∗
m−1
I ω (α) :=
ϕ∗y Im y
(ϕ−1
(M, ∂M )
y ) (λy .α) ∈ Ωc
with
y
dI ω (α) =
X
=
X
(ϕ−1 )∗ ω
ϕ∗y dIm y
∗
(ϕ−1
y ) (λy .α)
y
Z
∗
−1 ∗
)
(λ
.α)
−
(ϕ
)
ω
·
ϕ∗y (ϕ−1
y
y
y
Z
∗
(ϕ−1
)
(λ
.α)
=
α
−
ω
y
y
Qy
y
α.
M
The sum is finite since α
R has compact support. The change to an arbitrary form
ω̃ ∈ Ωm
(M
\
∂M
)
with
ω̃ = 1 is as above.
c
8. Cohomological interpretation of the Banyaga operator. For a connected
oriented manifold with corners M of dimension m (we assume that ∂M is not
empty) we consider the following diagram where only the dashed arrow
I ω does
R
m
not fit in commutingly. Here ω ∈ Ωc (M \ ∂M ) is a fixed form with ω = 1. All
instances of R in the diagram are connected by identities which fit commutingly
into the diagram. Each line is the definition of the corresponding top de Rham
cohomology space. The integral in the first line induces an isomorphism in cohomology since M \ ∂M is a connected oriented open manifold. The bottom triangle
commutes by Stokes’ theorem 5.
R
M \∂M
(M \ ∂M )
Ωm−1
c
_
(M,
)
Ωm−1
c
 _ ∂M
h
d
d
I
Ωm−1
(M )
c
/ Ωm
c (M \
 _ ∂M )
/ / Hcm (M \ ∂M )
/ Ωm
c (M, ∂M )
/ / Hcm (M, ∂M )
/ Ωm
c (M )
/ / Hcm (M )
(
R
6R
ω
d
R
M
0
R
R
∂1 M
◦j∂∗1 M
( R
M
R
m
m
Claim. M : Ωm
c (M, ∂M ) = Ωc (M ) → R
R induces
R Hc (M, ∂M ) = R.
R
m
Namely, given α, β ∈ Ωc (M, ∂M ) with α = β, we have α − dI ω (α) = ω M α
and similarly for β. This implies α − β = d (I ω (α) − I ω (β)) and hence [α] = [β] in
Hcm (M, ∂M ).
Claim. If M has non-empty boundary then Hcm (MR) = 0.
ω
For any form α ∈ Ωm
c (M ) we have α − dI (α) = ω M α, so α equals a multiple of
support. Now choose β ∈ Ωm−1
(M )R with
c
Rω modulo an exact form with compact
R
ω
∗
β
=
6
0.
Then
dβ
−
dI
(dβ)
=
ω M dβ
dβ
=
6
0;
for
example
with
j
M
∂1M ∂1M
shows that any multiple of ω is exact. Thus Hcm (M ) = 0.
9. Theorem (Poincaré–Lefschetz duality). For an oriented
connected manifold
R
with corners of dimension m the cohomological integral ∗ : Hcm (M, ∂M ) → R
8
MARTINS BRUVERIS, PETER W. MICHOR, ADAM PARUSIŃSKI, ARMIN RAINER
induces a non-degenerate bilinear form
k
PM
: H k (M ) × Hcm−k (M, ∂M ) → R
Z
Z
k
PM
([α], [β]) = [α] ∧ [β] =
α∧β.
∗
given by
M
This is in fact the special case for real coefficients of Lefschetz’ duality [9]. In
[14] Lefschetz duality is proven for piecewise linear stratified ∂-pseudomanifolds
in terms of intersection homology. Here we can give a proof based completely on
differential forms.
Proof. Note first that Ωc (M, ∂M ) is a graded ideal in Ω(M ), thus the integral
makes sense. The proof follows now, for example, [12, 12.14 – 12.16] with some
obvious changes.
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[14] G. Valette. A Lefschetz duality for intersection homology. Geom. Dedicata, 169:283–299, 2014.
[15] G. Valette. Stokes’ formula for stratified forms. Ann. Polon. Math., 114(3):197–206, 2015.
Martins Bruveris: Department of Mathematics, Brunel University London, Uxbridge,
UB8 3PH, United Kingdom
E-mail address: [email protected]
9
Peter W. Michor: Fakultät für Mathematik, Universität Wien, Oskar-MorgensternPlatz 1, A-1090 Wien, Austria.
E-mail address: [email protected]
Adam Parusiński: Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06108 Nice,
France
E-mail address: [email protected]
Armin Rainer: Fakultät für Mathematik, Universität Wien, Oskar-MorgensternPlatz 1, A-1090 Wien, Austria
E-mail address: [email protected]